Pythagorean-Hodograph (PH) Cubic Curves

A polynomial curve is a Pythagorean-Hodograph (PH) curve if is the square of another polynomial. The lowest-degree curves satisfying this condition are PH-cubics, here presented in Bézier form. The degrees of freedom of such a curve let you choose three of the four control polygon points. The remaining point is then determined.

THINGS TO TRY

SNAPSHOTS

DETAILS

Consider a polynomial parametric curve . By definition, its hodograph is its derivative . It is said to be Pythagorean if there is another polynomial such that . The curve is then said to have a Pythagorean hodograph, or, for short, to be a PH curve. it has the remarkable properties of having polynomial speed and permitting offset curves with rational parametrizations.

The lowest-degree curve allowing this property is three. Therefore this Demonstration shows cubic curves written in Bézier form, that is, represented by their control polygons (see [1]).

Denoting by the distance between and and by the angle , a cubic curve is PH if and only if and . This result allows the free choice of three of the control points; the fourth one is then determined.